Let $$M = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : x ^ { 2 } + y ^ { 2 } \leq r ^ { 2 } \right\} ,$$ where $r > 0$. Consider the geometric progression $a _ { n } = \frac { 1 } { 2 ^ { n - 1 } } , n = 1,2,3 , \ldots$. Let $S _ { 0 } = 0$ and, for $n \geq 1$, let $S _ { n }$ denote the sum of the first $n$ terms of this progression. For $n \geq 1$, let $C _ { n }$ denote the circle with center ( $S _ { n - 1 } , 0$ ) and radius $a _ { n }$, and $D _ { n }$ denote the circle with center ( $S _ { n - 1 } , S _ { n - 1 }$ ) and radius $a _ { n }$. Consider $M$ with $r = \frac { \left( 2 ^ { 199 } - 1 \right) \sqrt { 2 } } { 2 ^ { 198 } }$. The number of all those circles $D _ { n }$ that are inside $M$ is (A) 198 (B) 199 (C) 200 (D) 201
Let
$$M = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : x ^ { 2 } + y ^ { 2 } \leq r ^ { 2 } \right\} ,$$
where $r > 0$. Consider the geometric progression $a _ { n } = \frac { 1 } { 2 ^ { n - 1 } } , n = 1,2,3 , \ldots$. Let $S _ { 0 } = 0$ and, for $n \geq 1$, let $S _ { n }$ denote the sum of the first $n$ terms of this progression. For $n \geq 1$, let $C _ { n }$ denote the circle with center ( $S _ { n - 1 } , 0$ ) and radius $a _ { n }$, and $D _ { n }$ denote the circle with center ( $S _ { n - 1 } , S _ { n - 1 }$ ) and radius $a _ { n }$.\\
Consider $M$ with $r = \frac { \left( 2 ^ { 199 } - 1 \right) \sqrt { 2 } } { 2 ^ { 198 } }$. The number of all those circles $D _ { n }$ that are inside $M$ is\\
(A) 198\\
(B) 199\\
(C) 200\\
(D) 201